A straight-line drawing of a plane graph is called an open rectangle-ofinfluence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not necessarily a triangle. In this paper, we first obtain a sufficient condition for an inner tria...

The concept of linking was developed to produce Palais-Smale (PS) sequencesG(uk)→ a, G′(uk)→ 0 for C1functionals G that separate linking sets. These sequences produce critical points if they have convergent subsequences (i.e., ifG satisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations producemo...

Let R(∂) be a matrix whose entries are linear partial differential operators, with constant coefficients. Let Ω be a non-empty, open, bounded, convex set. We consider the homogeneous system R(∂)f = 0 in a neighborhood of Ω, subject to the boundary condition f = g in a neighborhood of ∂Ω. For a given smooth function g, we give a criterion for the (unique) existence of a smooth solution f to this...

the taguchi design of experiments was used to test the relative importance of medium components and environmental factors on poly(β-hydroxybutyrate)(phb) production by ralstonia eutropha. the optimum condition was obtained as: fructose concentration, 15 g/l; c/n ratio, 7.4; agitation speed 200 rpm; culture time, 40 h; temperature, 25 ° c; seed age, 15 h. at optimum condition the yield of phb pr...

The author has published a necessary and sufficient condition for a fmite loopless graph to have a spanning subgraph with a specified positive valency at each vertex (see [8,9). In the present paper it is contended that the condition can be made more useful as a tool of graph theory by imposing a maximality condition. 1. The condition for an I-factor Let G be a finite graph. Loops and multiple ...

In the context of list-coloring the vertices of a graph, Hall’s condition is a generalization of Hall’s Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph G with list assignment L satisfies Hall’s condition if for each subgraph H of G, the inequality |V (H)| 6∑σ∈C α(H(σ, L)) is satisfied, where C is the set of colors and α(H(σ, L)) is t...

Vizing's theorem states that the chromatic index 0 (G) of a graph G is either the maximum degree (G) or (G) + 1. A graph G is called overfull if jE(G)j > (G)bjV (G)j=2c. A suu-cient condition for 0 (G) = (G)+1 is that G contains an overfull subgraph H with (H) = (G). Plantholt proved that this condition is necessary for graphs with a universal vertex. In this paper, we conjecture that, for indi...

Given a directed graph G, the K node-disjoint paths problem consists in finding a partition of G into K node-disjoint paths, such that each path ends up in a given subset of nodes in G. This article provides a necessary condition for the K node-disjoint paths problem which combines (1) the structure of the reduced graph associated with G, (2) the structure of each strongly connected component o...

Consider a set of simple closed curves on a surface of genus g which fill the surface and which pairwise intersect at most once. We show that the asymptotic growth rate of the smallest number in such a set is 2 √ g as g → ∞. More generally, we give a precise asymptotic for filling sets of curves which pairwise intersect at most K ≥ 1 times. We then bound from below the cardinality of a filling ...

Let G be a connected graph of order n and X = {x ∈ V : d(x)≥n/2}. Suppose |X| ≥ 3 and G satisfies the modified Fan’s condition. We show that the vertices of the block B of G containing X form a cycle. This generalizes a result of Fan. We also give an efficient algorithm to obtain such a cycle. The complexity of this algorithm is O(n2). In case G is 2-connected, the condition |X| ≥ 3 can be remo...